# 'affine' Dialect

This dialect provides a powerful abstraction for affine operations and analyses.

## Polyhedral Structures ¶

MLIR uses techniques from polyhedral compilation to make dependence analysis and loop transformations efficient and reliable. This section introduces some of the core concepts that are used throughout the document.

### Dimensions and Symbols ¶

Dimensions and symbols are the two kinds of identifiers that can appear in the
polyhedral structures, and are always of
`index`

type.
Dimensions are declared in parentheses and symbols are declared in square
brackets.

Examples:

```
// A 2d to 3d affine mapping.
// d0/d1 are dimensions, s0 is a symbol
#affine_map2to3 = affine_map<(d0, d1)[s0] -> (d0, d1 + s0, d1 - s0)>
```

Dimensional identifiers correspond to the dimensions of the underlying structure being represented (a map, set, or more concretely a loop nest or a tensor); for example, a three-dimensional loop nest has three dimensional identifiers. Symbol identifiers represent an unknown quantity that can be treated as constant for a region of interest.

Dimensions and symbols are bound to SSA values by various operations in MLIR and use the same parenthesized vs square bracket list to distinguish the two.

Syntax:

```
// Uses of SSA values that are passed to dimensional identifiers.
dim-use-list ::= `(` ssa-use-list? `)`
// Uses of SSA values that are used to bind symbols.
symbol-use-list ::= `[` ssa-use-list? `]`
// Most things that bind SSA values bind dimensions and symbols.
dim-and-symbol-use-list ::= dim-use-list symbol-use-list?
```

SSA values bound to dimensions and symbols must always have ‘index’ type.

Example:

```
#affine_map2to3 = affine_map<(d0, d1)[s0] -> (d0, d1 + s0, d1 - s0)>
// Binds %N to the s0 symbol in affine_map2to3.
%x = memref.alloc()[%N] : memref<40x50xf32, #affine_map2to3>
```

### Restrictions on Dimensions and Symbols ¶

The affine dialect imposes certain restrictions on dimension and symbolic
identifiers to enable powerful analysis and transformation. An SSA value’s use
can be bound to a symbolic identifier if that SSA value is either 1. a region
argument for an op with trait `AffineScope`

(eg. `FuncOp`

), 2. a value defined
at the top level of an `AffineScope`

op (i.e., immediately enclosed by the
latter), 3. a value that dominates the `AffineScope`

op enclosing the value’s
use, 4. the result of a
constant operation, 5. the result of an
`affine.apply`

operation that recursively takes as
arguments any valid symbolic identifiers, or 6. the result of a
`dim`

operation on either a memref that
is an argument to a `AffineScope`

op or a memref where the corresponding
dimension is either static or a dynamic one in turn bound to a valid symbol.
*Note:* if the use of an SSA value is not contained in any op with the
`AffineScope`

trait, only the rules 4-6 can be applied.

Note that as a result of rule (3) above, symbol validity is sensitive to the
location of the SSA use. Dimensions may be bound not only to anything that a
symbol is bound to, but also to induction variables of enclosing
`affine.for`

and
`affine.parallel`

operations, and the result
of an
`affine.apply`

operation (which recursively
may use other dimensions and symbols).

### Affine Expressions ¶

Syntax:

```
affine-expr ::= `(` affine-expr `)`
| affine-expr `+` affine-expr
| affine-expr `-` affine-expr
| `-`? integer-literal `*` affine-expr
| affine-expr `ceildiv` integer-literal
| affine-expr `floordiv` integer-literal
| affine-expr `mod` integer-literal
| `-`affine-expr
| bare-id
| `-`? integer-literal
multi-dim-affine-expr ::= `(` `)`
| `(` affine-expr (`,` affine-expr)* `)`
```

`ceildiv`

is the ceiling function which maps the result of the division of its
first argument by its second argument to the smallest integer greater than or
equal to that result. `floordiv`

is a function which maps the result of the
division of its first argument by its second argument to the largest integer
less than or equal to that result. `mod`

is the modulo operation: since its
second argument is always positive, its results are always positive in our
usage. The `integer-literal`

operand for ceildiv, floordiv, and mod is always
expected to be positive. `bare-id`

is an identifier which must have type
index. The precedence of operations in an affine
expression are ordered from highest to lowest in the order: (1)
parenthesization, (2) negation, (3) modulo, multiplication, floordiv, and
ceildiv, and (4) addition and subtraction. All of these operators associate from
left to right.

A *multidimensional affine expression* is a comma separated list of
one-dimensional affine expressions, with the entire list enclosed in
parentheses.

**Context:** An affine function, informally, is a linear function plus a
constant. More formally, a function f defined on a vector $\vec{v} \in
\mathbb{Z}^n$ is a multidimensional affine function of $\vec{v}$ if $f(\vec{v})$
can be expressed in the form $M \vec{v} + \vec{c}$ where $M$ is a constant
matrix from $\mathbb{Z}^{m \times n}$ and $\vec{c}$ is a constant vector from
$\mathbb{Z}$. $m$ is the dimensionality of such an affine function. MLIR further
extends the definition of an affine function to allow ‘floordiv’, ‘ceildiv’, and
‘mod’ with respect to positive integer constants. Such extensions to affine
functions have often been referred to as quasi-affine functions by the
polyhedral compiler community. MLIR uses the term ‘affine map’ to refer to these
multidimensional quasi-affine functions. As examples, $(i+j+1, j)$, $(i \mod 2,
j+i)$, $(j, i/4, i \mod 4)$, $(2i+1, j)$ are two-dimensional affine functions of
$(i, j)$, but $(i \cdot j, i^2)$, $(i \mod j, i/j)$ are not affine functions of
$(i, j)$.

### Affine Maps ¶

Syntax:

```
affine-map-inline
::= dim-and-symbol-id-lists `->` multi-dim-affine-expr
```

The identifiers in the dimensions and symbols lists must be unique. These are the only identifiers that may appear in ‘multi-dim-affine-expr’. Affine maps with one or more symbols in its specification are known as “symbolic affine maps”, and those with no symbols as “non-symbolic affine maps”.

**Context:** Affine maps are mathematical functions that transform a list of
dimension indices and symbols into a list of results, with affine expressions
combining the indices and symbols. Affine maps distinguish between
indices and symbols because indices are inputs to the
affine map when the map is called (through an operation such as
affine.apply), whereas symbols are bound when the
map is established (e.g. when a memref is formed, establishing a memory
layout map).

Affine maps are used for various core structures in MLIR. The restrictions we impose on their form allows powerful analysis and transformation, while keeping the representation closed with respect to several operations of interest.

#### Named affine mappings ¶

Syntax:

```
affine-map-id ::= `#` suffix-id
// Definitions of affine maps are at the top of the file.
affine-map-def ::= affine-map-id `=` affine-map-inline
module-header-def ::= affine-map-def
// Uses of affine maps may use the inline form or the named form.
affine-map ::= affine-map-id | affine-map-inline
```

Affine mappings may be defined inline at the point of use, or may be hoisted to the top of the file and given a name with an affine map definition, and used by name.

Examples:

```
// Affine map out-of-line definition and usage example.
#affine_map42 = affine_map<(d0, d1)[s0] -> (d0, d0 + d1 + s0 floordiv 2)>
// Use an affine mapping definition in an alloc operation, binding the
// SSA value %N to the symbol s0.
%a = memref.alloc()[%N] : memref<4x4xf32, #affine_map42>
// Same thing with an inline affine mapping definition.
%b = memref.alloc()[%N] : memref<4x4xf32, affine_map<(d0, d1)[s0] -> (d0, d0 + d1 + s0 floordiv 2)>>
```

### Semi-affine maps ¶

Semi-affine maps are extensions of affine maps to allow multiplication,
`floordiv`

, `ceildiv`

, and `mod`

with respect to symbolic identifiers.
Semi-affine maps are thus a strict superset of affine maps.

Syntax of semi-affine expressions:

```
semi-affine-expr ::= `(` semi-affine-expr `)`
| semi-affine-expr `+` semi-affine-expr
| semi-affine-expr `-` semi-affine-expr
| symbol-or-const `*` semi-affine-expr
| semi-affine-expr `ceildiv` symbol-or-const
| semi-affine-expr `floordiv` symbol-or-const
| semi-affine-expr `mod` symbol-or-const
| bare-id
| `-`? integer-literal
symbol-or-const ::= `-`? integer-literal | symbol-id
multi-dim-semi-affine-expr ::= `(` semi-affine-expr (`,` semi-affine-expr)* `)`
```

The precedence and associativity of operations in the syntax above is the same as that for affine expressions.

Syntax of semi-affine maps:

```
semi-affine-map-inline
::= dim-and-symbol-id-lists `->` multi-dim-semi-affine-expr
```

Semi-affine maps may be defined inline at the point of use, or may be hoisted to the top of the file and given a name with a semi-affine map definition, and used by name.

```
semi-affine-map-id ::= `#` suffix-id
// Definitions of semi-affine maps are at the top of file.
semi-affine-map-def ::= semi-affine-map-id `=` semi-affine-map-inline
module-header-def ::= semi-affine-map-def
// Uses of semi-affine maps may use the inline form or the named form.
semi-affine-map ::= semi-affine-map-id | semi-affine-map-inline
```

### Integer Sets ¶

An integer set is a conjunction of affine constraints on a list of identifiers. The identifiers associated with the integer set are separated out into two classes: the set’s dimension identifiers, and the set’s symbolic identifiers. The set is viewed as being parametric on its symbolic identifiers. In the syntax, the list of set’s dimension identifiers are enclosed in parentheses while its symbols are enclosed in square brackets.

Syntax of affine constraints:

```
affine-constraint ::= affine-expr `>=` `affine-expr`
| affine-expr `<=` `affine-expr`
| affine-expr `==` `affine-expr`
affine-constraint-conjunction ::= affine-constraint (`,` affine-constraint)*
```

Integer sets may be defined inline at the point of use, or may be hoisted to the top of the file and given a name with an integer set definition, and used by name.

```
integer-set-id ::= `#` suffix-id
integer-set-inline
::= dim-and-symbol-id-lists `:` '(' affine-constraint-conjunction? ')'
// Declarations of integer sets are at the top of the file.
integer-set-decl ::= integer-set-id `=` integer-set-inline
// Uses of integer sets may use the inline form or the named form.
integer-set ::= integer-set-id | integer-set-inline
```

The dimensionality of an integer set is the number of identifiers appearing in dimension list of the set. The affine-constraint non-terminals appearing in the syntax above are only allowed to contain identifiers from dims and symbols. A set with no constraints is a set that is unbounded along all of the set’s dimensions.

Example:

```
// A example two-dimensional integer set with two symbols.
#set42 = affine_set<(d0, d1)[s0, s1]
: (d0 >= 0, -d0 + s0 - 1 >= 0, d1 >= 0, -d1 + s1 - 1 >= 0)>
// Inside a Region
affine.if #set42(%i, %j)[%M, %N] {
...
}
```

`d0`

and `d1`

correspond to dimensional identifiers of the set, while `s0`

and
`s1`

are symbol identifiers.

## Operations ¶

`affine.apply`

(mlir::AffineApplyOp) ¶

affine apply operation

The affine.apply operation applies an
affine mapping
to a list of SSA values, yielding a single SSA value. The number of
dimension and symbol arguments to `affine.apply`

must be equal to the
respective number of dimensional and symbolic inputs to the affine mapping;
the affine mapping has to be one-dimensional, and so the `affine.apply`

operation always returns one value. The input operands and result must all
have ‘index’ type.

Example:

```
#map10 = affine_map<(d0, d1) -> (d0 floordiv 8 + d1 floordiv 128)>
...
%1 = affine.apply #map10 (%s, %t)
// Inline example.
%2 = affine.apply affine_map<(i)[s0] -> (i+s0)> (%42)[%n]
```

Traits: AlwaysSpeculatableImplTrait

Interfaces: ConditionallySpeculatable, NoMemoryEffect (MemoryEffectOpInterface)

Effects: MemoryEffects::Effect{}

#### Attributes: ¶

Attribute | MLIR Type | Description |
---|---|---|

`map` | ::mlir::AffineMapAttr | AffineMap attribute |

#### Operands: ¶

Operand | Description |
---|---|

`mapOperands` | index |

#### Results: ¶

Result | Description |
---|---|

«unnamed» | index |

`affine.delinearize_index`

(mlir::AffineDelinearizeIndexOp) ¶

delinearize an index

Syntax:

```
operation ::= `affine.delinearize_index` $linear_index `into` ` ` `(` $basis `)` attr-dict `:` type($multi_index)
```

The `affine.delinearize_index`

operation takes a single index value and
calculates the multi-index according to the given basis.

Example:

```
%indices:3 = affine.delinearize_index %linear_index into (%c16, %c224, %c224) : index, index, index
```

In the above example, `%indices:3`

conceptually holds the following:

```
#map0 = affine_map<()[s0] -> (s0 floordiv 50176)>
#map1 = affine_map<()[s0] -> ((s0 mod 50176) floordiv 224)>
#map2 = affine_map<()[s0] -> (s0 mod 224)>
%indices_0 = affine.apply #map0()[%linear_index]
%indices_1 = affine.apply #map1()[%linear_index]
%indices_2 = affine.apply #map2()[%linear_index]
```

Traits: AlwaysSpeculatableImplTrait

Interfaces: ConditionallySpeculatable, NoMemoryEffect (MemoryEffectOpInterface)

Effects: MemoryEffects::Effect{}

#### Operands: ¶

Operand | Description |
---|---|

`linear_index` | index |

`basis` | index |

#### Results: ¶

Result | Description |
---|---|

`multi_index` | index |

`affine.for`

(mlir::AffineForOp) ¶

for operation

Syntax:

```
operation ::= `affine.for` ssa-id `=` lower-bound `to` upper-bound
(`step` integer-literal)? `{` op* `}`
lower-bound ::= `max`? affine-map-attribute dim-and-symbol-use-list | shorthand-bound
upper-bound ::= `min`? affine-map-attribute dim-and-symbol-use-list | shorthand-bound
shorthand-bound ::= ssa-id | `-`? integer-literal
```

The `affine.for`

operation represents an affine loop nest. It has one region
containing its body. This region must contain one block that terminates with
`affine.yield`

. *Note:* when
`affine.for`

is printed in custom format, the terminator is omitted. The
block has one argument of
`index`

type that
represents the induction variable of the loop.

The `affine.for`

operation executes its body a number of times iterating
from a lower bound to an upper bound by a stride. The stride, represented by
`step`

, is a positive constant integer which defaults to “1” if not present.
The lower and upper bounds specify a half-open range: the range includes the
lower bound but does not include the upper bound.

The lower and upper bounds of a `affine.for`

operation are represented as an
application of an affine mapping to a list of SSA values passed to the map.
The
same restrictions hold for
these SSA values as for all bindings of SSA values to dimensions and
symbols.

The affine mappings for the bounds may return multiple results, in which
case the `max`

/`min`

keywords are required (for the lower/upper bound
respectively), and the bound is the maximum/minimum of the returned values.
There is no semantic ambiguity, but MLIR syntax requires the use of these
keywords to make things more obvious to human readers.

Many upper and lower bounds are simple, so MLIR accepts two custom form
syntaxes: the form that accepts a single ‘ssa-id’ (e.g. `%N`

) is shorthand
for applying that SSA value to a function that maps a single symbol to
itself, e.g., `()[s]->(s)()[%N]`

. The integer literal form (e.g. `-42`

) is
shorthand for a nullary mapping function that returns the constant value
(e.g. `()->(-42)()`

).

Example showing reverse iteration of the inner loop:

```
#map57 = affine_map<(d0)[s0] -> (s0 - d0 - 1)>
func.func @simple_example(%A: memref<?x?xf32>, %B: memref<?x?xf32>) {
%N = dim %A, 0 : memref<?x?xf32>
affine.for %i = 0 to %N step 1 {
affine.for %j = 0 to %N { // implicitly steps by 1
%0 = affine.apply #map57(%j)[%N]
%tmp = call @F1(%A, %i, %0) : (memref<?x?xf32>, index, index)->(f32)
call @F2(%tmp, %B, %i, %0) : (f32, memref<?x?xf32>, index, index)->()
}
}
return
}
```

`affine.for`

can also operate on loop-carried variables (`iter_args`

) and
return the final values after loop termination. The initial values of the
variables are passed as additional SSA operands to the `affine.for`

following the operands for the loop’s lower and upper bounds. The
operation’s region has equivalent arguments for each variable representing
the value of the variable at the current iteration.

The region must terminate with an `affine.yield`

that passes all the current
iteration variables to the next iteration, or to the `affine.for`

’s results
if at the last iteration. For `affine.for`

’s that execute zero iterations, the
initial values of the loop-carried variables (corresponding to the SSA
operands) will be the op’s results.

For example, to sum-reduce a memref:

```
func.func @reduce(%buffer: memref<1024xf32>) -> (f32) {
// Initial sum set to 0.
%sum_0 = arith.constant 0.0 : f32
// iter_args binds initial values to the loop's region arguments.
%sum = affine.for %i = 0 to 10 step 2
iter_args(%sum_iter = %sum_0) -> (f32) {
%t = affine.load %buffer[%i] : memref<1024xf32>
%sum_next = arith.addf %sum_iter, %t : f32
// Yield current iteration sum to next iteration %sum_iter or to %sum
// if final iteration.
affine.yield %sum_next : f32
}
return %sum : f32
}
```

```
%res:2 = affine.for %i = 0 to 128 iter_args(%arg0 = %init0, %arg1 = %init1)
-> (index, index) {
%y0 = arith.addi %arg0, %c1 : index
%y1 = arith.addi %arg1, %c2 : index
affine.yield %y0, %y1 : index, index
}
```

If the `affine.for`

defines any values, a yield terminator must be
explicitly present. The number and types of the “affine.for” results must
match the initial values in the `iter_args`

binding and the yield operands.

Traits: AutomaticAllocationScope, ImplicitAffineTerminator, RecursiveMemoryEffects

Interfaces: ConditionallySpeculatable, LoopLikeOpInterface, RegionBranchOpInterface

#### Operands: ¶

Operand | Description |
---|---|

«unnamed» | any type |

#### Results: ¶

Result | Description |
---|---|

`results` | any type |

`affine.if`

(mlir::AffineIfOp) ¶

if-then-else operation

Syntax:

```
operation ::= `affine.if` if-op-cond `{` op* `}` (`else` `{` op* `}`)?
if-op-cond ::= integer-set-attr dim-and-symbol-use-list
```

The `affine.if`

operation restricts execution to a subset of the loop
iteration space defined by an integer set (a conjunction of affine
constraints). A single `affine.if`

may end with an optional `else`

clause.

The condition of the `affine.if`

is represented by an
integer set (a conjunction of affine constraints),
and the SSA values bound to the dimensions and symbols in the integer set.
The
same restrictions hold for
these SSA values as for all bindings of SSA values to dimensions and
symbols.

The `affine.if`

operation contains two regions for the “then” and “else”
clauses. `affine.if`

may return results that are defined in its regions.
The values defined are determined by which execution path is taken. Each
region of the `affine.if`

must contain a single block with no arguments,
and be terminated by `affine.yield`

. If `affine.if`

defines no values,
the `affine.yield`

can be left out, and will be inserted implicitly.
Otherwise, it must be explicit. If no values are defined, the else block
may be empty (i.e. contain no blocks).

Example:

```
#set = affine_set<(d0, d1)[s0]: (d0 - 10 >= 0, s0 - d0 - 9 >= 0,
d1 - 10 >= 0, s0 - d1 - 9 >= 0)>
func.func @reduced_domain_example(%A, %X, %N) : (memref<10xi32>, i32, i32) {
affine.for %i = 0 to %N {
affine.for %j = 0 to %N {
%0 = affine.apply #map42(%j)
%tmp = call @S1(%X, %i, %0)
affine.if #set(%i, %j)[%N] {
%1 = affine.apply #map43(%i, %j)
call @S2(%tmp, %A, %i, %1)
}
}
}
return
}
```

Example with an explicit yield (initialization with edge padding):

```
#interior = affine_set<(i, j) : (i - 1 >= 0, j - 1 >= 0, 10 - i >= 0, 10 - j >= 0)> (%i, %j)
func.func @pad_edges(%I : memref<10x10xf32>) -> (memref<12x12xf32) {
%O = alloc memref<12x12xf32>
affine.parallel (%i, %j) = (0, 0) to (12, 12) {
%1 = affine.if #interior (%i, %j) {
%2 = load %I[%i - 1, %j - 1] : memref<10x10xf32>
affine.yield %2
} else {
%2 = arith.constant 0.0 : f32
affine.yield %2 : f32
}
affine.store %1, %O[%i, %j] : memref<12x12xf32>
}
return %O
}
```

Traits: ImplicitAffineTerminator, NoRegionArguments, RecursiveMemoryEffects, RecursivelySpeculatableImplTrait

Interfaces: ConditionallySpeculatable, RegionBranchOpInterface

#### Operands: ¶

Operand | Description |
---|---|

«unnamed» | any type |

#### Results: ¶

Result | Description |
---|---|

`results` | any type |

`affine.load`

(mlir::AffineLoadOp) ¶

affine load operation

The “affine.load” op reads an element from a memref, where the index for each memref dimension is an affine expression of loop induction variables and symbols. The output of ‘affine.load’ is a new value with the same type as the elements of the memref. An affine expression of loop IVs and symbols must be specified for each dimension of the memref. The keyword ‘symbol’ can be used to indicate SSA identifiers which are symbolic.

Example 1:

```
%1 = affine.load %0[%i0 + 3, %i1 + 7] : memref<100x100xf32>
```

Example 2: Uses ‘symbol’ keyword for symbols ‘%n’ and ‘%m’.

```
%1 = affine.load %0[%i0 + symbol(%n), %i1 + symbol(%m)] : memref<100x100xf32>
```

Traits: MemRefsNormalizable

Interfaces: AffineMapAccessInterface, AffineReadOpInterface

#### Operands: ¶

Operand | Description |
---|---|

`memref` | memref of any type values |

`indices` | index |

#### Results: ¶

Result | Description |
---|---|

`result` | any type |

`affine.max`

(mlir::AffineMaxOp) ¶

max operation

The “max” operation computes the maximum value result from a multi-result affine map.

Example:

```
%0 = affine.max (d0) -> (1000, d0 + 512) (%i0) : index
```

Traits: AlwaysSpeculatableImplTrait

Interfaces: ConditionallySpeculatable, NoMemoryEffect (MemoryEffectOpInterface)

Effects: MemoryEffects::Effect{}

#### Attributes: ¶

Attribute | MLIR Type | Description |
---|---|---|

`map` | ::mlir::AffineMapAttr | AffineMap attribute |

#### Operands: ¶

Operand | Description |
---|---|

`operands` | index |

#### Results: ¶

Result | Description |
---|---|

«unnamed» | index |

`affine.min`

(mlir::AffineMinOp) ¶

min operation

Syntax:

```
operation ::= ssa-id `=` `affine.min` affine-map-attribute dim-and-symbol-use-list
```

The `affine.min`

operation applies an
affine mapping
to a list of SSA values, and returns the minimum value of all result
expressions. The number of dimension and symbol arguments to `affine.min`

must be equal to the respective number of dimensional and symbolic inputs to
the affine mapping; the `affine.min`

operation always returns one value. The
input operands and result must all have ‘index’ type.

Example:

```
%0 = affine.min affine_map<(d0)[s0] -> (1000, d0 + 512, s0)> (%arg0)[%arg1]
```

Traits: AlwaysSpeculatableImplTrait

Interfaces: ConditionallySpeculatable, NoMemoryEffect (MemoryEffectOpInterface)

Effects: MemoryEffects::Effect{}

#### Attributes: ¶

Attribute | MLIR Type | Description |
---|---|---|

`map` | ::mlir::AffineMapAttr | AffineMap attribute |

#### Operands: ¶

Operand | Description |
---|---|

`operands` | index |

#### Results: ¶

Result | Description |
---|---|

«unnamed» | index |

`affine.parallel`

(mlir::AffineParallelOp) ¶

multi-index parallel band operation

The “affine.parallel” operation represents a hyper-rectangular affine parallel band, defining zero or more SSA values for its induction variables. It has one region capturing the parallel band body. The induction variables are represented as arguments of this region. These SSA values always have type index, which is the size of the machine word. The strides, represented by steps, are positive constant integers which defaults to “1” if not present. The lower and upper bounds specify a half-open range: the range includes the lower bound but does not include the upper bound. The body region must contain exactly one block that terminates with “affine.yield”.

The lower and upper bounds of a parallel operation are represented as an application of an affine mapping to a list of SSA values passed to the map. The same restrictions hold for these SSA values as for all bindings of SSA values to dimensions and symbols. The list of expressions in each map is interpreted according to the respective bounds group attribute. If a single expression belongs to the group, then the result of this expression is taken as a lower(upper) bound of the corresponding loop induction variable. If multiple expressions belong to the group, then the lower(upper) bound is the max(min) of these values obtained from these expressions. The loop band has as many loops as elements in the group bounds attributes.

Each value yielded by affine.yield will be accumulated/reduced via one of the reduction methods defined in the AtomicRMWKind enum. The order of reduction is unspecified, and lowering may produce any valid ordering. Loops with a 0 trip count will produce as a result the identity value associated with each reduction (i.e. 0.0 for addf, 1.0 for mulf). Assign reductions for loops with a trip count != 1 produces undefined results.

Note: Calling AffineParallelOp::build will create the required region and block, and insert the required terminator if it is trivial (i.e. no values are yielded). Parsing will also create the required region, block, and terminator, even when they are missing from the textual representation.

Example (3x3 valid convolution):

```
func.func @conv_2d(%D : memref<100x100xf32>, %K : memref<3x3xf32>) -> (memref<98x98xf32>) {
%O = alloc memref<98x98xf32>
affine.parallel (%x, %y) = (0, 0) to (98, 98) {
%0 = affine.parallel (%kx, %ky) = (0, 0) to (2, 2) reduce ("addf") {
%1 = affine.load %D[%x + %kx, %y + %ky] : memref<100x100xf32>
%2 = affine.load %K[%kx, %ky] : memref<3x3xf32>
%3 = arith.mulf %1, %2 : f32
affine.yield %3 : f32
}
affine.store %0, O[%x, %y] : memref<98x98xf32>
}
return %O
}
```

Example (tiling by potentially imperfectly dividing sizes):

```
affine.parallel (%ii, %jj) = (0, 0) to (%N, %M) step (32, 32) {
affine.parallel (%i, %j) = (%ii, %jj)
to (min(%ii + 32, %N), min(%jj + 32, %M)) {
call @f(%i, %j) : (index, index) -> ()
}
}
```

Traits: AutomaticAllocationScope, ImplicitAffineTerminator, MemRefsNormalizable, RecursiveMemoryEffects, RecursivelySpeculatableImplTrait

Interfaces: ConditionallySpeculatable, LoopLikeOpInterface

#### Attributes: ¶

Attribute | MLIR Type | Description |
---|---|---|

`reductions` | ::mlir::ArrayAttr | Reduction ops |

`lowerBoundsMap` | ::mlir::AffineMapAttr | AffineMap attribute |

`lowerBoundsGroups` | ::mlir::DenseIntElementsAttr | 32-bit signless integer elements attribute |

`upperBoundsMap` | ::mlir::AffineMapAttr | AffineMap attribute |

`upperBoundsGroups` | ::mlir::DenseIntElementsAttr | 32-bit signless integer elements attribute |

`steps` | ::mlir::ArrayAttr | 64-bit integer array attribute |

#### Operands: ¶

Operand | Description |
---|---|

`mapOperands` | index |

#### Results: ¶

Result | Description |
---|---|

`results` | any type |

`affine.prefetch`

(mlir::AffinePrefetchOp) ¶

affine prefetch operation

The “affine.prefetch” op prefetches data from a memref location described with an affine subscript similar to affine.load, and has three attributes: a read/write specifier, a locality hint, and a cache type specifier as shown below:

```
affine.prefetch %0[%i, %j + 5], read, locality<3>, data : memref<400x400xi32>
```

The read/write specifier is either ‘read’ or ‘write’, the locality hint specifier ranges from locality<0> (no locality) to locality<3> (extremely local keep in cache). The cache type specifier is either ‘data’ or ‘instr’ and specifies whether the prefetch is performed on data cache or on instruction cache.

Interfaces: AffineMapAccessInterface

#### Attributes: ¶

Attribute | MLIR Type | Description |
---|---|---|

`isWrite` | ::mlir::BoolAttr | bool attribute |

`localityHint` | ::mlir::IntegerAttr | 32-bit signless integer attribute whose minimum value is 0 whose maximum value is 3 |

`isDataCache` | ::mlir::BoolAttr | bool attribute |

#### Operands: ¶

Operand | Description |
---|---|

`memref` | memref of any type values |

`indices` | index |

`affine.store`

(mlir::AffineStoreOp) ¶

affine store operation

The “affine.store” op writes an element to a memref, where the index for each memref dimension is an affine expression of loop induction variables and symbols. The ‘affine.store’ op stores a new value which is the same type as the elements of the memref. An affine expression of loop IVs and symbols must be specified for each dimension of the memref. The keyword ‘symbol’ can be used to indicate SSA identifiers which are symbolic.

Example 1:

```
affine.store %v0, %0[%i0 + 3, %i1 + 7] : memref<100x100xf32>
```

Example 2: Uses ‘symbol’ keyword for symbols ‘%n’ and ‘%m’.

```
affine.store %v0, %0[%i0 + symbol(%n), %i1 + symbol(%m)] : memref<100x100xf32>
```

Traits: MemRefsNormalizable

Interfaces: AffineMapAccessInterface, AffineWriteOpInterface

#### Operands: ¶

Operand | Description |
---|---|

`value` | any type |

`memref` | memref of any type values |

`indices` | index |

`affine.vector_load`

(mlir::AffineVectorLoadOp) ¶

affine vector load operation

The “affine.vector_load” is the vector counterpart of affine.load. It reads a slice from a MemRef, supplied as its first operand, into a vector of the same base elemental type. The index for each memref dimension is an affine expression of loop induction variables and symbols. These indices determine the start position of the read within the memref. The shape of the return vector type determines the shape of the slice read from the memref. This slice is contiguous along the respective dimensions of the shape. Strided vector loads will be supported in the future. An affine expression of loop IVs and symbols must be specified for each dimension of the memref. The keyword ‘symbol’ can be used to indicate SSA identifiers which are symbolic.

Example 1: 8-wide f32 vector load.

```
%1 = affine.vector_load %0[%i0 + 3, %i1 + 7] : memref<100x100xf32>, vector<8xf32>
```

Example 2: 4-wide f32 vector load. Uses ‘symbol’ keyword for symbols ‘%n’ and ‘%m’.

```
%1 = affine.vector_load %0[%i0 + symbol(%n), %i1 + symbol(%m)] : memref<100x100xf32>, vector<4xf32>
```

Example 3: 2-dim f32 vector load.

```
%1 = affine.vector_load %0[%i0, %i1] : memref<100x100xf32>, vector<2x8xf32>
```

TODOs:

- Add support for strided vector loads.
- Consider adding a permutation map to permute the slice that is read from memory (see vector.transfer_read).

Traits: MemRefsNormalizable

Interfaces: AffineMapAccessInterface, AffineReadOpInterface

#### Operands: ¶

Operand | Description |
---|---|

`memref` | memref of any type values |

`indices` | index |

#### Results: ¶

Result | Description |
---|---|

`result` | vector of any type values |

`affine.vector_store`

(mlir::AffineVectorStoreOp) ¶

affine vector store operation

The “affine.vector_store” is the vector counterpart of affine.store. It writes a vector, supplied as its first operand, into a slice within a MemRef of the same base elemental type, supplied as its second operand. The index for each memref dimension is an affine expression of loop induction variables and symbols. These indices determine the start position of the write within the memref. The shape of th input vector determines the shape of the slice written to the memref. This slice is contiguous along the respective dimensions of the shape. Strided vector stores will be supported in the future. An affine expression of loop IVs and symbols must be specified for each dimension of the memref. The keyword ‘symbol’ can be used to indicate SSA identifiers which are symbolic.

Example 1: 8-wide f32 vector store.

```
affine.vector_store %v0, %0[%i0 + 3, %i1 + 7] : memref<100x100xf32>, vector<8xf32>
```

Example 2: 4-wide f32 vector store. Uses ‘symbol’ keyword for symbols ‘%n’ and ‘%m’.

```
affine.vector_store %v0, %0[%i0 + symbol(%n), %i1 + symbol(%m)] : memref<100x100xf32>, vector<4xf32>
```

Example 3: 2-dim f32 vector store.

```
affine.vector_store %v0, %0[%i0, %i1] : memref<100x100xf32>, vector<2x8xf32>
```

TODOs:

- Add support for strided vector stores.
- Consider adding a permutation map to permute the slice that is written to memory (see vector.transfer_write).

Traits: MemRefsNormalizable

Interfaces: AffineMapAccessInterface, AffineWriteOpInterface

#### Operands: ¶

Operand | Description |
---|---|

`value` | vector of any type values |

`memref` | memref of any type values |

`indices` | index |

`affine.yield`

(mlir::AffineYieldOp) ¶

Yield values to parent operation

Syntax:

```
operation ::= `affine.yield` attr-dict ($operands^ `:` type($operands))?
```

“affine.yield” yields zero or more SSA values from an affine op region and terminates the region. The semantics of how the values yielded are used is defined by the parent operation. If “affine.yield” has any operands, the operands must match the parent operation’s results. If the parent operation defines no values, then the “affine.yield” may be left out in the custom syntax and the builders will insert one implicitly. Otherwise, it has to be present in the syntax to indicate which values are yielded.

```
Traits: AlwaysSpeculatableImplTrait, MemRefsNormalizable, ReturnLike, Terminator
Interfaces: ConditionallySpeculatable, NoMemoryEffect (MemoryEffectOpInterface)
Effects: MemoryEffects::Effect{}
#### Operands:
| Operand | Description |
| :-----: | ----------- |
| `operands` | any type
### 'affine.load' operation
Syntax:
```

operation ::= ssa-id `=`

`affine.load`

ssa-use `[`

multi-dim-affine-map-of-ssa-ids `]`

`:`

memref-type

```
The `affine.load` op reads an element from a memref, where the index for each
memref dimension is an affine expression of loop induction variables and
symbols. The output of 'affine.load' is a new value with the same type as the
elements of the memref. An affine expression of loop IVs and symbols must be
specified for each dimension of the memref. The keyword 'symbol' can be used to
indicate SSA identifiers which are symbolic.
Example:
```mlir
Example 1:
%1 = affine.load %0[%i0 + 3, %i1 + 7] : memref<100x100xf32>
Example 2: Uses 'symbol' keyword for symbols '%n' and '%m'.
%1 = affine.load %0[%i0 + symbol(%n), %i1 + symbol(%m)]
: memref<100x100xf32>
```

### ‘affine.store’ operation ¶

Syntax:

```
operation ::= ssa-id `=` `affine.store` ssa-use, ssa-use `[` multi-dim-affine-map-of-ssa-ids `]` `:` memref-type
```

The `affine.store`

op writes an element to a memref, where the index for each
memref dimension is an affine expression of loop induction variables and
symbols. The ‘affine.store’ op stores a new value which is the same type as the
elements of the memref. An affine expression of loop IVs and symbols must be
specified for each dimension of the memref. The keyword ‘symbol’ can be used to
indicate SSA identifiers which are symbolic.

Example:

```
Example 1:
affine.store %v0, %0[%i0 + 3, %i1 + 7] : memref<100x100xf32>
Example 2: Uses 'symbol' keyword for symbols '%n' and '%m'.
affine.store %v0, %0[%i0 + symbol(%n), %i1 + symbol(%m)]
: memref<100x100xf32>
```

### ‘affine.dma_start’ operation ¶

Syntax:

```
operation ::= `affine.dma_Start` ssa-use `[` multi-dim-affine-map-of-ssa-ids `]`, `[` multi-dim-affine-map-of-ssa-ids `]`, `[` multi-dim-affine-map-of-ssa-ids `]`, ssa-use `:` memref-type
```

The `affine.dma_start`

op starts a non-blocking DMA operation that transfers
data from a source memref to a destination memref. The source and destination
memref need not be of the same dimensionality, but need to have the same
elemental type. The operands include the source and destination memref’s each
followed by its indices, size of the data transfer in terms of the number of
elements (of the elemental type of the memref), a tag memref with its indices,
and optionally at the end, a stride and a number_of_elements_per_stride
arguments. The tag location is used by an AffineDmaWaitOp to check for
completion. The indices of the source memref, destination memref, and the tag
memref have the same restrictions as any affine.load/store. In particular, index
for each memref dimension must be an affine expression of loop induction
variables and symbols. The optional stride arguments should be of ‘index’ type,
and specify a stride for the slower memory space (memory space with a lower
memory space id), transferring chunks of number_of_elements_per_stride every
stride until %num_elements are transferred. Either both or no stride arguments
should be specified. The value of ‘num_elements’ must be a multiple of
‘number_of_elements_per_stride’.

Example:

```
For example, a DmaStartOp operation that transfers 256 elements of a memref
'%src' in memory space 0 at indices [%i + 3, %j] to memref '%dst' in memory
space 1 at indices [%k + 7, %l], would be specified as follows:
%num_elements = arith.constant 256
%idx = arith.constant 0 : index
%tag = memref.alloc() : memref<1xi32, 4>
affine.dma_start %src[%i + 3, %j], %dst[%k + 7, %l], %tag[%idx],
%num_elements :
memref<40x128xf32, 0>, memref<2x1024xf32, 1>, memref<1xi32, 2>
If %stride and %num_elt_per_stride are specified, the DMA is expected to
transfer %num_elt_per_stride elements every %stride elements apart from
memory space 0 until %num_elements are transferred.
affine.dma_start %src[%i, %j], %dst[%k, %l], %tag[%idx], %num_elements,
%stride, %num_elt_per_stride : ...
```

### ‘affine.dma_wait’ operation ¶

Syntax:

```
operation ::= `affine.dma_Start` ssa-use `[` multi-dim-affine-map-of-ssa-ids `]`, `[` multi-dim-affine-map-of-ssa-ids `]`, `[` multi-dim-affine-map-of-ssa-ids `]`, ssa-use `:` memref-type
```

The `affine.dma_start`

op blocks until the completion of a DMA operation
associated with the tag element ‘%tag[%index]’. %tag is a memref, and %index has
to be an index with the same restrictions as any load/store index. In
particular, index for each memref dimension must be an affine expression of loop
induction variables and symbols. %num_elements is the number of elements
associated with the DMA operation. For example:

Example:

```
affine.dma_start %src[%i, %j], %dst[%k, %l], %tag[%index], %num_elements :
memref<2048xf32, 0>, memref<256xf32, 1>, memref<1xi32, 2>
...
...
affine.dma_wait %tag[%index], %num_elements : memref<1xi32, 2>
```